Calculus Question

Question

1. Suppose f is continuous on [0,8] and the only solutions of the equation f(x)=2 are x=1 and x=7. If f(3)=-14, explain why f(5)<2

Answer 1

By assumptions we know that f(5) can not be equal to 2. If f(5)>2 and since f(3)=-14 we know from the intermediate value theorem that there must be a value c in (3,5) such that f(c)=2. But this contradicts the assumption that the only solutions of f(x)=2 are x=1 and x=7. Hence, the only option for the value f(5) is strictly less than 2 which finishes the proof.